Theorem zfrepclf 1477

Description: An inference rule based on the Axiom of Replacement. Typically, φ defines a function from x to y.

Hypotheses

Ref	Expression
zfrepclf.1	⊢ (w ∈ A → ∀x w ∈ A)
zfrepclf.2	⊢ A ∈ V
zfrepclf.3	⊢ (x ∈ A → ∃z∀y(φ → y = z))

Assertion

Ref	Expression
zfrepclf	⊢ ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ∧ φ))

Distinct variable group(s):   y,z,A   φ,z   w,A   x,y,z   x,w

Proof of Theorem zfrepclf

Step	Hyp	Ref	Expression
1		zfrepclf.2	. 2 ⊢ A ∈ V
2		ax-17 925	. . . . . 6 ⊢ (w ∈ v → ∀x w ∈ v)
3		zfrepclf.1	. . . . . 6 ⊢ (w ∈ A → ∀x w ∈ A)
4	2, 3	hbeq 1171	. . . . 5 ⊢ (v = A → ∀x v = A)
5		eleq2 1150	. . . . . 6 ⊢ (v = A → (x ∈ v ↔ x ∈ A))
6		zfrepclf.3	. . . . . 6 ⊢ (x ∈ A → ∃z∀y(φ → y = z))
7	5, 6	syl6bi 187	. . . . 5 ⊢ (v = A → (x ∈ v → ∃z∀y(φ → y = z)))
8	4, 7	19.21ai 740	. . . 4 ⊢ (v = A → ∀x(x ∈ v → ∃z∀y(φ → y = z)))
9		ax-17 925	. . . . 5 ⊢ (φ → ∀zφ)
10	9	zfrep3 1476	. . . 4 ⊢ (∀x(x ∈ v → ∃z∀y(φ → y = z)) → ∃z∀y(y ∈ z ↔ ∃x(x ∈ v ∧ φ)))
11	8, 10	syl 12	. . 3 ⊢ (v = A → ∃z∀y(y ∈ z ↔ ∃x(x ∈ v ∧ φ)))
12	5	anbi1d 469	. . . . . . 7 ⊢ (v = A → ((x ∈ v ∧ φ) ↔ (x ∈ A ∧ φ)))
13	4, 12	biexd 783	. . . . . 6 ⊢ (v = A → (∃x(x ∈ v ∧ φ) ↔ ∃x(x ∈ A ∧ φ)))
14	13	bibi2d 470	. . . . 5 ⊢ (v = A → ((y ∈ z ↔ ∃x(x ∈ v ∧ φ)) ↔ (y ∈ z ↔ ∃x(x ∈ A ∧ φ))))
15	14	bialdv 935	. . . 4 ⊢ (v = A → (∀y(y ∈ z ↔ ∃x(x ∈ v ∧ φ)) ↔ ∀y(y ∈ z ↔ ∃x(x ∈ A ∧ φ))))
16	15	biexdv 936	. . 3 ⊢ (v = A → (∃z∀y(y ∈ z ↔ ∃x(x ∈ v ∧ φ)) ↔ ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ∧ φ))))
17	11, 16	mpbid 170	. 2 ⊢ (v = A → ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ∧ φ)))
18	1, 17	vtocle 1391	1 ⊢ ∃z∀y(y ∈ z ↔ ∃x(x ∈ A ∧ φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  zfrep3cl 1478  zfrep4 1479

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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